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Some remarks on the theory of surface waves of finite amplitude. (English. Russian original) Zbl 0641.76007
Sov. Math., Dokl. 35, No. 3, 599-603 (1987); translation from Dokl. Akad. Nauk SSSR 294, No. 5, 1033-1037 (1987).
We consider the plane potential motion of a fluid in a gravitational field. We shall study steady-state waves on the surface of a fluid of infinite depth. Although this wave problem has been considered many times before this, the structure of the set of solutions and the question of bifurcation of wave motions remain so far unsolved. Recently papers have appeared devoted to the bifurcation problem [e.g.: M. S. Longuet- Higgins, J. Fluid Mech. 151, 457-475 (1985; Zbl 0575.76025)] in which this question was investigated numerically. In spite of the beautiful and intuitively convincing results, they are nevertheless devoid of demonstrative force. At the same time the method of “proving” computations, applied with success in a number of papers of the author and colleagues [e.g.: the author and V. Yu. Petrovitch, Dokl. Akad. Nauk SSSR 277, 265-269 (1984; Zbl 0596.65039)], makes it possible in wave theory to obtain a number of exact results not accessible by analytic and topological methods. In this note we present investigations needed to construct a numerical algorithm and carry out further analysis.

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q30 Navier-Stokes equations